The sum of several numbers divided by their count nicely summarizes the overall size of the numbers involved in a single value.

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4
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2answers
177 views

Basic question about natural density

Suppose that we have a sequence of finite sets $A_1, A_2, \ldots$, which partition $\mathbb{N}$. I am making no other assumptions on the $A_n$ - i.e. there could be any amount of interleaving between ...
4
votes
5answers
275 views

what is the argument for not using the average of an average?

I want to disprove someone's calculation of percentage of cash sales for a year by taking summing percentage of cash sales by month and dividing by 12. I sense the correct way is to take total cash ...
4
votes
1answer
5k views

Average Distance Between Random Points in a Rectangle

My question is similar to this one but for rectangles instead of lines. Suppose I have a rectangle with sides of length $L_w$ and $L_h$. What is the average distance between two uniformly-distributed ...
4
votes
2answers
131 views

Average $lcm(a,b)$, $ 1\le a \le b \le n$, and asymptotic behavior

What is the average value for $\mathrm{lcm}(a,b)$, with $ 1\le a \le b \le n$, for a given $n$, and what is the asymptotic behavior? The $\mathrm{lcm}$ is the least common multiple. I have ...
4
votes
3answers
7k views

Calculate average wind direction

What is the best way to average wind direction? I have found many conflicting suggestions elsewhere. Best one I saw is to average SINs of all angles in radians and take inverse SIN of the result. ...
4
votes
1answer
40 views

What is the name of this type of “average”?

Given two variables $x$ and $y$, $$\frac{2}{1/x + 1/y} $$ looks like some kind of average of $x$ and $y$. Is there a name for it? Where can i look up more information (such as properties) of this ...
4
votes
1answer
157 views

Average Number of Roots of a Polynomial modulo p

Let $f \in \mathbb{Z}[X]$ be an irreducible non-constant polynomial, and consider this polynomial modulo $p$ for each prime $p$. On average, how many roots does $f$ have modulo $p$? I.e., if $r(p)$ ...
4
votes
1answer
135 views

Dirichlet Series and Average Values of Certain Arithmetic Functions

If an arithmetic function $f(n)$ has Dirichlet series $\zeta(s) \prod_{i,j = 1} \frac{\zeta(a_i s)}{\zeta(b_j s)}$, for which values of $a_{i}$ and $b_{j}$ is the following true? That \begin{align} ...
4
votes
1answer
217 views

Power Mean Random Distribution

I'm trying to find a the distribution for the power mean of $n$ random variables on $[0,1]$. I've got the arithmetic mean: $\frac{n}{(n-1)!}\sum_{k=0}^{\lfloor ...
4
votes
1answer
133 views

Average distance to a random point in a rectangle from an arbitrary point

I'm interested in the mean distance between an arbitrary 2D point, $(p, q)$, and a uniformly distributed point inside a rectangle defined by the lower left and upper right vertices $(x_0, y_0)$ and ...
4
votes
1answer
110 views

Notation for Average of a Set?

In particular, I have some set $S = \{s_1, s_2, s_3, ..., s_n\}$ and a subset $S^\prime$, and I want to denote the average of the elements in $S^\prime$. I would generally just use ...
4
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1answer
4k views

Real world examples for Mode in Statistics

I am looking for some real world examples for mode in Statistics involving topics which students like say Football or Social networks. Also they need to clearly identify differences in the usefulness ...
4
votes
1answer
125 views

Probability value in a skewed data

A question goes like this: Given that there 2 questions out of 3 questions to pass an exam and follow are details of number of people who attended the questions correctly, what is the maximum number ...
4
votes
2answers
43 views

Maximise difference between mean and median

I have a set of numbers and I want to maximise the difference between the mean and the median by removing a given number of elements. For instance I have the following set: ...
4
votes
3answers
57 views

Convergence and Absolute Convergence of Arithmetic Mean of a sequence

Suppose $\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n |x_i|$ exists. Does $\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n x_i$ exist? How about the converse? My thoughts: I guess for the sequence ...
4
votes
1answer
144 views

Mean value of the rotation angle is 126.5°

In the paper "Applications of Quaternions to Computation with Rotations" by Eugene Salamin, 1979 (click here), they get 126.5 degrees as the mean value of the rotation angle of a random rotation (by ...
4
votes
0answers
437 views

Average transformation matrix?

I have several estimates of the transformation matrix between two planes and some values that give some indication of the error involved in the estimate. How can I use this information to gain the ...
3
votes
4answers
183 views

If $\sum\limits_{k=1}^n y_k\geq n$ and $\sum\limits_{k=1}^n \frac{1}{y_k}\geq n$, then $\prod\limits_{k=1}^n y_k\geq 1$?

Let $y_1,\ldots y_n$ be positive real numbers satisfying $y_1+\cdots+y_n\geq n$ and $\displaystyle{\frac{1}{y_1}+\cdots+\frac{1}{y_n}\geq n}$. Is it true that $y_1y_2\cdots y_n\geq 1$?
3
votes
2answers
111 views

In what sense is the “average” of $sin(x)$ equal to 0?

I think it makes intuitive sense that if a phenomenon is described by simple sinusoidal oscillation, it is "on average" equal to its midrange, but I think I failed in trying to make that statement ...
3
votes
3answers
250 views

Definition of random

Suppose that you has to guess given a set of numbers If they are random. The mathematical expectation Is there a definition of randomness that allow this prove/test? Is even possible? if so: How ...
3
votes
2answers
103 views

An upper bound for an average exponentiated weighted sum of a vector from $\{-1,1\}^n$

Suppose $\mathcal{S}=\{\mathbf{x}:\mathbf{x}\in\{-1,1\}^n\}$, that is, $\mathcal{S}$ contains all $2^n$ vectors of length $n$ containing -1 and 1. I am interested in the following average: ...
3
votes
3answers
106 views

The average value of the function $y=\tan(2x)$ over the interval $[0,\frac{\pi}{8}]$

I was given the following question in a technology free exam. How would one go about solving this without the use of a calculator? NB. I am currently in my last year of high school so please don't ...
3
votes
1answer
37 views

Is this some known mathematical concept?

I was thinking about a way to do a "weighted average" (that's what I call it, could be dead wrong) of a variable $x$ defined for a given range $x_1\leq x\leq x_N$, weighted by an always positive ...
3
votes
2answers
868 views

How to find the average value of $y = e^x$ between $x = e$ and $x = 2e$? [closed]

What approach would be ideal in finding the average value of $y = e^x$ between $x = e$ and $x = 2e$?
3
votes
3answers
54 views

Why this statistical equation is true?

Pardon my ignorance. I also don't know how to properly show the equation. So I attached URL of latex... Why this equation is true for any real number $k$, when $\overline{X}$ is average of individual ...
3
votes
1answer
58 views

What is the most elementary proof of these inequalities?

Let $p$ be a non-zero integer, and let $x_1$, $\ldots$, $x_n$ be $n$ positive real numbers. Then we define the $p$-th power mean $M_p$ of these numbers as $$ M_p \colon= (\frac{x_1^p + \ldots + ...
3
votes
1answer
498 views

The Exponential decay.

I am studying semiconductor physics. there is a paragraph about Drude model in E.spenke's book "Electronic semiconductors" page 259 in art §9: "if on the average, a time $τ$ elapses between two ...
3
votes
1answer
70 views

Is it always true that “max $\ge$ average + sigma”?

Assume that $i$ from $1,\ldots,N$, $x_i \ge 0$ and: $$\mathrm{avg} = \frac{\sum_i x_i}{N}$$ $$\sigma = \sqrt\frac{\sum_i{(x_i-\mathrm{avg})^2}}{N}$$ Is that true that: $$\max_i x_i \ge ...
3
votes
2answers
2k views

APR Calculation

I'm hoping someone can clarify this for me. The model/example is this: We lend an amount of 1498.50 (loan amount). Other fees total 39.95. The term of the loan is for 12 months. There is no ...
3
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2answers
58 views

How do I properly average these percentages?

I have attendance records for an annual event: person 1: $4$ of $8$ attended $= 50.00$%, or every $2$ years person 2: $1$ of $4$ attended $= 25.00$%, or every $4$ years ...
3
votes
1answer
80 views

What about $\lim_{n\to\infty}\frac{\sum_{k=1}^n s_k\mu(k)}{n}$, for the zeros of Dirichlet eta function $s_k=1+\frac{2\pi k}{\log 2}i$ with $k\geq 1$?

Let for integers $k\geq 1$ the corresponding zeros of Dirichlet eta function of the form $$s_k=1+\frac{2\pi k}{\log 2}i,$$ then we can consider the following puzzle, when we multiply previous ...
3
votes
2answers
118 views

Very fascinating probability game about maximising greed?

Two people play a mathematical game. Each person chooses a number between 1 and 100 inclusive, with both numbers revealed at the same time. The person who has a smaller number will keep their number ...
3
votes
2answers
285 views

Expressing a summation using matrix algebra

Consider the $r \times n$ matrix $$\begin{pmatrix} X_{11} & X_{12} & \cdots & X_{1n} \\ X_{21} & X_{22} & \cdots & X_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ ...
3
votes
1answer
48 views

Is the norm of the average $\le$ the norm of the max?

Given $\pmb X \in \mathcal{R}^p$, denote the elements of $\pmb X$ as $\pmb x_i$ for $i= 1, \dots, n$. Denote the $t(\pmb X)$ as the average of $\pmb X$ \begin{equation} \pmb t(\pmb X) = \frac 1 n ...
3
votes
1answer
135 views

How do I average a new number into an established percentage if I know the number of items?

This seems like something I should be able to do but I can not for the life of me figure it out. I'm writing a program to calculate an average score; let's say that my data looks like this. x = ...
3
votes
2answers
324 views

Course Grade - Statistics Question

I'm confused about finding the standard deviation of various weighted objects in a problem. For example, consider an academic setting, with four assessments as follows: ...
3
votes
1answer
2k views

leading and lagging moving average indicator

What are leading short and lagging long moving average indicators and how do we calculate them? e.g. for the following data set, and let window size be 2. Can you ...
3
votes
4answers
80 views

Determine drug concentration over time, given its halflife and dosage

I want to calculate which of two doses is going to have the most active ingredient over the total time of an experiment. So as an example let's say I have a drug which has a halflife of 5 hours, and ...
3
votes
3answers
3k views

what does `ensemble average` mean?

I'm studying this paper and somewhere in the conclusion part is written: "Since this rotation of the coherency matrix is carried out based on the ensemble average of polarimetric scattering ...
3
votes
1answer
66 views

What is the maximum value of the sum $\sum_{i=1}^L(\bar{x}-x_i)$, in this specific case.

Let $x_i$ be a positive real variable, with $i=1,2,...,K$. We denote by $\bar{x}$ the average value of the values $x_1, x_2,...,x_K$. Let $a=\min_i x_i$ and $b=\max_i x_i$, then $x_i \in [a,b]$. My ...
3
votes
1answer
88 views

What is this average called?

What is this average called? I tried Googling it but couldn't find anything about it. It's something between an arithmetic mean and a geometric mean, by which I mean you iterate both averages for two ...
3
votes
1answer
259 views

How to find the function $f$ that satisfies $f(x, y) = f(x^{-1}, y^{-1})^{-1}$ and $f(x, y)$ is $\approx$ $average(x, y)$?

Fist of all, I'm a programmer, not a mathematician, and I'm sorry for my non native English. And I'm sorry if the question is not appropriate, it is my first time here. Or if the question has no ...
3
votes
1answer
29 views

Simplified Averages - Always true?

While working on an "average rating" function for a website, I came across an idea for simplified averages, but I want to confirm that it always works. If it does, I might be able to keep a running ...
3
votes
2answers
966 views

What do curly brackets mean in this formula?

In this paper, in the Formula at the beginning of 2.2, we have $B=\{b_i(O_t)\}$ where $i=0,1$ - the number of probability formula $O_t$ - the state at moment $t$ $b_i(O_t)$ - two probabilities ...
3
votes
1answer
48 views

Expected value (mean) of function from polyline

Suppose we have a polyline that has such properties: It consists of n segments First segment's ends are (0, 0) and ...
3
votes
0answers
121 views

Finding a summarizing vector for average angle calculation

Let $L$ and $R$ be two bags of positive vectors such that all vectors have length $k$. Define the distance $d_{avg}$ between the bags as the average pairwise angle between the vectors. Is is possible ...
3
votes
0answers
55 views

Find spikes in data

I have some datasets and I need to find spikes in them. Imagine the data looks like trading data. If the spike is big enough, I need to log it, otherwise, proceed in the analysis. I tried with a ...
2
votes
3answers
116 views

Problem with calculating average.

I came across the following question. A man travels a distance of $20$ miles at $60$ miles/hr and then return over the same route at $40$ miles/hr. What is the average rate for the round trip in ...
2
votes
3answers
207 views

Average amount of money received from pulling three random coins

Let's say I have a bag of coins, which contains 1 quarter, 2 dimes, 3 nickles and 4 pennies. If I were to randomly pull out 3 coins, on average, how much money would I get?
2
votes
3answers
74 views

What is the mean score for the $20$ rolls?

A fair die is rolled twenty times. The results are shown in the bar graph. What is the mean score for the $20$ rolls?